OK,
After thinking a lot about it, I got the solution
\int_0^{y_i}\int_0^{\eta} f(x,\xi) \, d\xi \, d\eta
\, = \,
\sum\limits_{s=1}^j\sum\limits_{r=1}^{s-1} (y_s-y_{s-1}) \, \int_{\eta_{r-1}}^{\eta_r}f(x,\xi) \, d\xi\
\, + \,
\sum\limits_{s=1}^j \int_{y_{s-1}}^{y_s}...